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New action for simplicial gravityCurvature and relation to Regge calculus
Wolfgang WielandInstitute for Gravitation and the Cosmos, Penn State
Tux 320 February 2015
What is the semi-classical limit for all spins?
LQG boundary states twisted geometries
spinfoam amplitudes Regge geometries GR
dynamics for twisted geometries
~→ 0
j →∞, βj =const.
~→ 0
continuumlimit
~→ 0
For large spins (large distances) and small Barbero–Immirzi parameter weseem to get the Regge action. What do we get for small spins—shortdistances—high energies?
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Basic idea
The semi-classical limit of covariant LQG for arbitrary values of j and βshould give a theory of discretized gravity in terms of Ashtekar–Barberovariables. Can we find such a classical theory without knowing the exactquantum theory behind?
A spinfoam model assigns amplitudes Ae, Af ,Av ,... to the elementary building blocks of thesimplicial complex.In the semi-classical limit these amplitudes turninto action functionals:Ae ∝ eiSe , Af ∝ eiSf , Av ∝ eiSv ,...The desired action will be of the form:Sspinfoam =
∑e Se +
∑f Sf +
∑v Sv + ....
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Outline of the talk
Main message: Covariant LQG suggests a new action for simplicial gravity withspinors as the fundamental configuration variables: The theory has aHamiltonian and local gauge symmetries. Generic solutions represent twistedgeometries, but the solution space contains also Regge configurations.Table of contents1 Worldline action for simplicial gravity2 Regge solutions, and on-shell Regge action3 Conclusion
References:*ww, New action for simplicial gravity in four dimensions, Class. Quant. Grav. 32 (2015), arXiv:1407.0025.*ww, One-dimensional action for simplicial gravity in three dimensions, Phys. Rev. D 90 (2014), arXiv:1402.6708.*ww, Twistorial phase space for complex Ashtekar variables, Class. Quant. Grav. 29 (2012), arXiv:1107.5002.*L Freidel and S Speziale, From twistors to twisted geometries, Phys. Rev. D 82 (2010), arXiv:1006.0199.
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Worldline action for simplicial gravity
Plebański principleThe BF action is topological, and determines the symplectic structure ofthe theory:
SBF[Σ, A] =
∫M
1
2`P2
(∗ Σαβ −
1
βΣαβ
)︸ ︷︷ ︸
Παβ
∧Fαβ [A]. (1)
General relativity follows from the simplicity constraints added to theaction:Σαβ ∧ Σµν ∝ εαβµν . (2)
With the solutions:Σαβ =
{±eα ∧ eβ ,± ∗ (eα ∧ eβ).
(3)Notation:
α, β, γ . . . are internal Lorentz indices.Σαβ is an so(1, 3)-valued two-form.Aαβ is an SO(1, 3) connection, with Fαβ = dAαβ +Aαµ ∧A
µβ denoting its curvature.
eα is the tetrad, diagonalizing the four-dimensional metric g = ηαβeα ⊗ eβ .
`P2 = 8π/G, and β is the Barbero–Immirzi parameter, ~ = 1 = c.
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Simplicial discretization
Faust: Faust:Das Pentagramma macht dir Pein? The pentagram prohibits thee?Ei sage mir, du Sohn der Hölle, Why, tell me now, thou Son of Hades,Wenn das dich bannt, wie kamst du denn herein? If that prevents, how cam’st thou in to me?Wie ward ein solcher Geist betrogen? Could such a spirit be so cheated?Mephistopheles: Mephistopheles:Beschaut es recht! es ist nicht gut gezogen: Inspect the thing: the drawing’s not completed.Der eine Winkel, der nach außen zu, The outer angle, you may see,Ist, wie du siehst, ein wenig offen. Is open left—the lines don’t fit it.
Aus Goethes Faust.7 / 32
Construction of the action, 1/2Step 1: Discretize the action:
SBF[Σ, A] =
∫M
Παβ ∧ Fαβ ≈∑f :faces
∫τf
Παβ
∫f
Fαβ ≡∑f :faces
Sf .
Step 2: Employ the non-Abelian Stoke’s theorem:Ff =
∫f
h−1Fh =
∮∂f
h−1Dh.
Step 3: Define the smeared flux:Πf =
∫τf
h−1Πh.
Step 4: Contract the two formulae in a commonframe:Sf = −
∮∂f
dt[h−1γt(1)
D
dthγt(1)
]αβ
Παβf (t).
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Construction of the action, 2/2Step 5: Introduce spinors to diagonalize both holonomies and fluxes:
Παβf (t) =
1
2εABω
(Af (t)π
B)f (t) + cc.,
[hγt]AB
= Pexp(−∫γt
A)A
B=ω˜Af (t)πfB(t)− π˜Af (t)ωfB(t)√
Ef (t)√E˜ f (t)
.
We also need the area-matching constraint:∆f := π˜fAω˜Af − πfAωAf ≡ E˜ f (t)− Ef (t).
Putting the pieces together yields the face action:Sf [Z,Z˜ , A, ζ] =
=
∮∂f
dt[πA
D
dtωA − π˜A d
dtω˜A − ζ∆
]+ cc. (5)
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Geometric interpretation of the spinorsWe define the complex null tetrad
`α = iπAπA, kα = iωAωA,
1
2
(xα + iyα
)= mα = iωAπA, mα = iπAωA.
Area, and plane of the triangle:(β + i) Ar = β`P
2πAωA, Σαβ ∝ x[αyβ]. (6)
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Geometric interpretation of the spinorsWe define the complex null tetrad
`α = iπAπA, kα = iωAωA,
1
2
(xα + iyα
)= mα = iωAπA, mα = iπAωA.
The area-matching constraint ∆ generates the transformations:πA → ezπA, ωA → e−zωA, z ∈ C. (7)
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Geometric interpretation of the spinorsWe define the complex null tetrad
`α = iπAπA, kα = iωAωA,
1
2
(xα + iyα
)= mα = iωAπA, mα = iπAωA.
Rotations around the z-axis:πA → e−
iϕ2 πA, ωA → e
iϕ2 ωA, ϕ ∈ [0, 4π). (8)
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Geometric interpretation of the spinorsWe define the complex null tetrad
`α = iπAπA, kα = iωAωA,
1
2
(xα + iyα
)= mα = iωAπA, mα = iπAωA.
Boosts into the z-direction:πA → e−
ξ2 πA, ωA → e
ξ2ωA, ξ ∈ R. (9)
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Linear simplicity constraintsInstead of discretizing the quadratic simplicity constraints
Σαβ ∧ Σµν ∝ εαβµν , (10)we will use the linear simplicity constraints:For a tetrahedron Te (dual to an edge e) there exist an internal future-orientedfour-vector nαe such that the fluxes through the four bounding triangles τf(dual to a face f : e ⊂ ∂f ) annihilate nαe :∫
τf
Σαβnβe = 0. (11)
The spinorial parametrization turns the simplicity constraints into thefollowing complex conditions:Vf =
i
β + iπfAω
Af + cc.
!= 0, (12a)
Wef = nAAe πfAωf
A
!= 0. (12b)
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Adding the simplicity constraints
The simplicity constraints reduce the SO(1, 3) spin connection Aαβ tothe SU(2)n Asthekar–Barbero connection:Aα = nµ
[1
2εµν
αρAνρ + βAαµ
]. (13)
We introduce Lagrange multipliers λ ∈ R and z ∈ C and get thefollowing constrained action for each face in the discretization:Sface[Z,Z˜ |ζ, z, λ|A, n] =
∮∂f
(πADωA − π˜Adω˜A − ζ(π˜Aω˜A − πAωA)+
− λ
2
( i
β + iπAω
A + cc.)− z nAAπAωA
)+ cc., (14)
where DπA = dπA +AατABαπB is the SU(2)n covariant differential.Problem: There is no term in the action that would determine thet-dependence of the normal nαe along the edges e(t).We now have to make a proposal.
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Four-dimensional closure constraintAny proposal for the dynamics of the time normals must respect theclosure constraint at the vertices (four-simplices):We define the four-momenta:
peα = g neαVol(e). (15)At every four simplex we have the closure constraint:∑
outgoing edges eat v
peα =∑
incoming edges eat v
peα. (16)
The constant g is dimensionful:[g] =
[Λ
8πG
]=
mass
volume(17)
Remarks:Vol(e) ∝ 2
9nαεαβµνL1
βL2µL
3ν , with e.g.: L1
α = −τABαωf1A π
f1B + cc.
In a locally flat geometry, the closure constraint follows from the vanishing of torsion:Tα
= Deα
= 0⇒1
3!εαρµµD
(eρ ∧ eµeν
)= 0⇒
∑edges e
at v
±epeα = 0.
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Dynamics of the time-normals
Any proposal for the dynamics of the time-normals- must respect the four-dimensional closure constraint, and- be consistent with all symmetries of the action.The following action fulfills these requirements:
Sedge[X, p|N,Vol(e)] =
∫e
(pαdXα − N
2
(g−1 pαp
α + gVol2(e))). (18)
We just need an additional boundary term at the vertices:Svertex[Yv, {Xev}e3v, {vev}e3v] =
∑e:e3v
(Y αv −Xα
ev
)vevα . (19)
Where N is a Lagrange multiplier imposing the mass-shell condition:C :=
1
2
(g−1 pαp
α + gVol2(e))
!= 0. (20)
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Putting the pieces together – defining the actionAdding the face, edge and vertex contributions gives us a proposal for anaction for discretized gravity in first-order variables:
Sspinfoam =∑f :faces
Sface[Zf , Z˜f ∣∣ζf , zf , λf ∣∣A∂f , n∂f ]+
+∑e:edges
Sedge[Xe, pe
∣∣Ne,Vol(e)]+
+∑
v:vertices
Svertex
[Yv, {Xev}e3v, {vev}e3v
]. (21)
Notation:Zf and Z˜f are the twistors Zf : ∂f → T ' C4 parametrizing the SL(2,C)holonomy-flux variables.ζf , λf and zf are Lagrange multipliers imposing the area-matching constraint andsimplicity constraints respectively.A is the SU(2)n Ashtekar–Barbero connection along the edges of the discretization.n denotes the time normal of the elementary tetrahedra.pe is the volume-weighted time-normal, of the tetrahedron dual to the edge e.Vol(e) denotes the corresponding three-volume.N is a Lagrange multiplier imposing the mass-shell condition C = 0.
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Simplicial gravity as a theory of interacting particles?
A dictionary:spinfoam formalism – auxiliary particlestetrahedra – particlesfour-simplices – interaction verticesthree-volume – masstetrahedral shapes – internal SU(2) DOFtorsion=0 – conservation of four-momentum
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Immediate question
Is this a reasonable model for discretized gravity?1 The Equations of motion generate twisted geometries: Every trianglehas a unique area, but the shape of a triangle depends on whether wecompute it from the metric in one adjacent four-simplex or the other.
2 Relation to Regge calculus: We can restrict ourselves to Regge-likesolutions.
3 The model has curvature: There is a deficit angle once we go around atriangle.
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Regge solutions
Hamiltonian formulationThe Hamiltonian:H = AαGα +
∑f :∂f⊃e
(ζf∆f + ζf ∆f + zfWef + zfWef + λfVf
)+NC, (22)
generates the t-evolution along the edges of the discretization:d
dtωAf =
{H,ωAf
}. (23)
The fundamental Poisson brackets are:{pα, X
β} = δβα,{πfA, ω
Bf ′}
= +δff ′δBA ,{
π˜fA, ω˜Bf ′} = −δff ′δBA .
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Dirac anaysis and reduced Hamiltonian
H = AαGα +∑
f :∂f⊃e
(ζf∆f + ζf ∆f + zfWef + zfWef + λfVf
)+NC. (24)
- The Hamiltonian preserves all constraints for zf = 0.- TheWef simplicity constraint is second class, all other constraints first class.- There are no secondary constraints.- ζf = 0 wlog.
H = AαGα +∑
f :∂f⊃e
λfVf +NC. (25)Relevant constraints of the system:
mass-shell: C = 12
(g−1pαp
α + gVol2)
first-class simplicity: V = iβ+iπAω
A + cc.
Gauß constraint: Gα = −∑f:∂f⊂e L
fα generates SU(2)n transformations,
with Lfα = −τABαπfAω
fB + cc., and su(2)n generators τα : [τα, τβ ] = ε µ
αβ τν .
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Action of the first-class simplicity constraintsThe first-class simplicity constraints V = 0 generates a four-screw:{
V, ωA}
=i
β + iωA,
{V, πA
}= − i
β + iπA.
A combination of a rotation and a boost preserving the triangle’s plane:
*F Hellmann and W Kamiński, Holonomy spin foam models: Asymptotic geometry of the partition function , JHEP 1310 (2013), arXiv:1307.1679.*V Bonzom, Spin foam models for quantum gravity from lattice path integrals, Phys. Rev D. 80 (2009), arXiv:0905.1501.21 / 32
Action of the first-class simplicity constraintsThe first-class simplicity constraints V = 0 generates a four-screw:{
V, ωA}
=i
β + iωA,
{V, πA
}= − i
β + iπA.
A combination of a rotation and a boost preserving the triangle’s plane:
*F Hellmann and W Kamiński, Holonomy spin foam models: Asymptotic geometry of the partition function , JHEP 1310 (2013), arXiv:1307.1679.*V Bonzom, Spin foam models for quantum gravity from lattice path integrals, Phys. Rev D. 80 (2009), arXiv:0905.1501.21 / 32
Action of the mass-shell conditionVolume flow in the shape space of all tetrahedra with fixed areas:
The simplicity constraintsimpose that the Σαβ fluxesdefine planes in internalMinkowski space.The Gauß constraintGα tellsus that these planes close toform a tetrahedron.The mass-shell conditiongenerates a globale rotationplus a shear plus a U(1)rotation in the plane of everytriangle.
C =1
2
(g−1pαp
α+ gVol
2), with: Vol
2 ∝2
9nαε
αβµνL
1βL
2µL
3ν .
*E Bianchi and HM Haggard, Bohr-Sommerfeld Quantization of Space, Phys.Rev. D 86 (2012), arXiv:1208.2228. 22 / 32
Restriction to stable tetrahedra
We restrict ourselves to tetrahedra that are stable under the volumeflow:
The mass shell condition generates residual U(1) transformation ofthe spinors in the triangles:πA → e−
iϕ2 πA, ωA → e+ iϕ
2 ωA. (26)We can tune these angles in such a way that they cancel theunwanted rotation from the simplicity constraints.We are then left with a pure boost.
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Gauge fixingIt works as follows (for e.g. the spinors in the 1-plane):
We align the dyade in the 1-plane with the(34)-edge,and require that the Hamiltonian preservesthis additional condition.
H = AαGα +
4∑f=1
λfVf +NC. (27)
This fixes the Lagrange multiplier λf in terms of N :λ1(N) =
g`P2
4(1 + β2)
Vol2
sin2 ϑ12
Ar2 + Ar1 cosϑ12
Ar1 Ar2N (28)
There are three possible choices to align the spinors to an edge. Theresult (28) is independent of this ambiguity.Remark:ϑIJ is the dihedral angle, Vol denotes the volume and ArI is the area of the I-th triangle.
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Effective equations of motion
Our alignment brings the evolution equations into a simple form:∇dtωAf =
d
dtωAf + ΓABω
Bf = +
ξf2ωAf , (29a)
∇dtπAf =
d
dtπAf + ΓABπ
Bf = −ξf
2πAf , (29b)
Geometric interpretation:Γ ∈ su(2)n is the Levi-Civita connection along the edges.ξf ∈ Rmeasures the extrinsic curvature in the face f .The parallel transport around the face is a pure boost:
[hf ]AB =
[Pexp
(−∮∂f
dtΓ
)]AB
=
=1
πCωC
[e−
12
∮∂f dtξωAπB + e+ 1
2
∮∂f dtξπAωB
] (30)
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Deficit angle and extrinsic curvature
Inter-tetrahedral angles:cosh Ξvf = −ηµνneµne
′ν , with: e ∩ e′ = v, and: e, e′ ⊂ ∂f. (31)
Deficit angle around a triangle:Ξf :=
∑v: vertices in f
Ξvf =
∮∂f
dtξ. (32)
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The on-shell action is the Regge actionWe evaluate the on-shell action for volume-stable geometries.
The action consists of a symplectic potential plus constraints.The constraints vanish on-shell.The symplectic potential pαdXα does not contribute either, because of:∫
epαdXα EOM
= pα[Xα∣∣e(1)−Xα
∣∣e(0)
], and
∑edges e
at v
±epeα = 0 (33)
All contributions come from the symplectic potential for the spinors.Regge action at the fixed point of the volume flow
Sspinfoam[π, ω, p,X]EOM
=∑f :faces
∮∂fπfAdωAf + cc.
EOM=
1
2
∑f :faces
πfAωAf Ξf + cc. =
EOM=
1
2
∑f :faces
(β + i)
β`P2
Arf Ξf + cc. =1
8πG
∑f :faces
Arf Ξf = SRegge[{`b}b:bones
].
Solutions of the equations of motion extremize the spinfoam action,hence they also bring the Regge action to an extremum.27 / 32
Conclusion
Summary
General picture: The simplicial edges turn into the worldlines of a system ofauxiliary particles scattering in a flat auxiliary manifold. Every tetrahedron carries aconserved four-momentum. Its norm is not mass but volume.Results:
Generic solutions represent twisted geometries that are theboundary data of loop quantum gravity.Regge configurations appear at the fixed point of the volume flow,where the on-shell action Sspinfoam turns into the Regge action SRegge.
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Open issues of the model
Role of local Lorentz invariance: The addition of the pαdXα termbreaks local SL(2,C) invariance down to the little group SU(2)n.What is the physical role of the Xα-background geometry with flatMinkowski metric ηαβ? Is there a relation to teleparallelism?We have only shown local existence of Regge-like solutions. Openproblem: Find explicit solutions that triangulate physical (Ricci flat)spacetime geometries.
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Relevance for loop quantum gravity
Quantum kinematics (simple problem): The instantaneous Hilbertspace is the Hilbert space of projected spin network functions.Area↔ norm of the Pauli–Lubanski vector. Quantization of area↔quantization of spin in the auxiliary particle model.Quantum dynamics (hard problem): Take the spinfoam action andreformulate loop quantum gravity as a one-dimensional QFT overthe edges of the discretization. The spinfoam amplitudes turn intothe S-matrix amplitudes of an auxiliary worldline model.
*S Alexandrov and ER Livine , SU(2) loop quantum gravity seen from covariant theory , Phys. Rev. D 67 (2003), arXiv:gr-qc/0209105.
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Thank you for your attention
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